$$
\begin{aligned}
P(H \mid E) & =\frac{P(E \mid H) \cdot P(H)}{P(E)} \\
p(\boldsymbol{X} \mid \alpha) & =\int_{\theta} p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha) \mathrm{d} \theta \\
p(\theta \mid \boldsymbol{X}, \alpha) & =\frac{p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha)}{p(\boldsymbol{X} \mid \alpha)} \propto p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha)
\end{aligned}
$$

\[
\begin{array}{l}
\lim \limits_{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sin \left(t^{2}\right) d t}{x^{6}} \quad \frac{0}{0} \text { 型 } \\
=\lim \limits_{x \rightarrow 0} \frac{\sin x^{4} \cdot 2 x}{6 x^{5}} \quad \text { 此步的求导流程 } \varphi(x)=\int_{0}^{x^{2}} \sin \left(t^{2}\right) d t \\
\varphi^{\prime}(x)=\sin \left(\left(x^{2}\right)^{2}\right) \cdot\left(x^{2}\right)^{\prime}=\sin x^{4} \cdot 2 x
\\
=\lim \limits_{x \rightarrow 0} \frac{\sin x^{4}}{3 x^{4}}
\end{array}
\]

一元二次方程 $a x^{2}+b x+c=D(a \neq 0)$ 求根公式: 
\[X=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\left(b^{2}-4 a c \geqslant 0\right)\]

根的判别式: 当 $\Delta>0$ 时，一开二次方程有唡个不相等的实数根当 $\Delta=0$ 时，一元二次方程有两个相等的实数根当 $\Delta<0$ 时, 一元二次方程无实数根

根与系数的关系: 
\[x_{1}+x_{2}=-\frac{b}{a}, x_{1} x_{2}=\frac{c}{a}\]

一次丞数 ![math](https://open.ocrmath.com/uploads/inset/a20a/a20aace7d4d5e5864d6fa5bd7083fc34_4901.jpg)